Optimization processes are techniques that achieve the best outcome in systems with functional constraints, such as constrained resources and combinations of the competitive resources. Examples of these constraints can include, among others, the power that can be output from a generator and the power that can be transmitted by a transmission line. The best outcome may be defined as the outcome that results in the lowest value of an objective function, which is typically presented mathematically in terms of cost. For example, the lowest value may represent the lowest price paid for generation of electrical power. In another example, the best outcome may be defined as the outcome that results in the highest value of an objective function. For example, the highest value may represent the highest price paid for generation of electrical power. The outcome of such an optimization process is the best schedule of resource values (the optimal point) that satisfy the constraints. For example, the best schedule of resource values may define a power generation schedule for several power generators.
Typically, the optimal point is based upon some important system parameters, such as the value (e.g. price) of the resource or the cost of producing the resource. The optimization process reaches an equilibrium point where the incremental value of each available resource (not on a limit) is equal to the calculated incremental value of the resource given the set functional constraints on limit. This equilibrium point is termed the Karush-Kuhn-Tucker (KKT) condition in mathematics and the Nash equilibrium point in economics. Resources establishing the equilibrium may be called marginal resources.
The calculated measure of value (e.g. price) of each resource at the optimal point is derived from the benefit of relaxing each binding functional constraint. This measure of the value is called the shadow value. In some embodiments, the shadow value may be a shadow price. The Lagrangian multiplier may also be used to represent the worth for purposes of optimization.
In the simplest form of the solution of the optimization process, the functional constraints that are not on limit have no worth (their shadow values are zero) whereas all binding functional constraints on limits have a positive value and thus provide value. The values of the binding functional constraints are used to calculate the replacement value (clearing value) of resources that are fully utilized (above the margin) or completely unutilized (below the margin).
Occasionally the solution of the optimization process results in some non-binding but fully loaded (NBFL) constraints. These are functional constraints that are on limits at the solution but have shadow values of zero. This condition indicates that shadow values are indeterminate as well as the derivative values calculated for resources above or below the margin. The worth of these resources becomes nebulous as their resulting values depend upon the order of the processing of constraints (path-dependency) during the solution.